ROM SAF Report 14 Ref: SAF/ROM/METO/REP/RSR/014 Web: www.romsaf.org Date: 16 Oct 2013 ROM SAF Report 14 A review of the geodesy calculations in ROPP Ian Culverwell Met Office, Exeter, UK Culverwell: ROPP geodesy review ROM SAF Report 14 Document Author Table Name Function Date Comments Prepared by: I. D. Culverwell ROM SAF Project Team 16 Oct 2013 Reviewed by: D. Offiler ROM SAF Project Team 27 Feb 2013 Reviewed by: J. R. Eyre ROM SAF Steering Group Member 13 May 2013 Approved by: K. B. Lauritsen ROM SAF Project Manager 26 Sep 2013 Document Change Record Issue/Revision Date By Description 1st draft 25 Feb 2013 IDC First draft, based on report attached to ROPP ticket 159 2nd draft 27 Mar 2013 IDC Corrections following 1st review by DO Version 1.0 11 Sep 2013 IDC Amendments after contacting ROPP-2.0 beta reviewer ROM SAF The Radio Occultation Meteorology Satellite Application Facility (ROM SAF) is a decentralised processing centre under EUMETSAT which is responsible for operational processing of GRAS radio occultation data from the Metop satellites and RO data from other missions. The ROM SAF delivers bending angle, refractivity, temperature, pressure, and humidity profiles in near-real time and offline for NWP and climate users. The offline profiles are further processed into climate products consisting of gridded monthly zonal means of bending angle, refractivity, temperature, humidity, and geopotential heights together with error descriptions. The ROM SAF also maintains the Radio Occultation Processing Package (ROPP) which contains software modules that will aid users wishing to process, quality-control and assimilate radio occulta- tion data from any radio occultation mission into NWP and other models. The ROM SAF Leading Entity is the Danish Meteorological Institute (DMI), with Cooperating Entities: i) European Centre for Medium-Range Weather Forecasts (ECMWF) in Reading, United Kingdom, ii) Institut D’Estudis Espacials de Catalunya (IEEC) in Barcelona, Spain, and iii) Met Office in Exeter, United Kingdom. To get access to our products or to read more about the project please go to: http://www.romsaf.org Intellectual Property Rights All intellectual property rights of the ROM SAF products belong to EUMETSAT. The use of these products is granted to every interested user, free of charge. If you wish to use these products, EU- METSAT’s copyright credit must be shown by displaying the words "copyright (year) EUMETSAT" on each of the products used. ROM SAF Report 14 Culverwell: ROPP geodesy review Abstract This report reviews the geodesy calculations in ROPP,in response to concerns raised by the ROPP2.0 beta reviewer in 2008. Various expressions for the surface gravity, effective radius, radius of curva- ture and geopotential height are compared, and the impact of the differences in RO applications is assessed. Supporting sensitivity studies are reported. A simple model is developed that sheds light on some of the findings. 3 Culverwell: ROPP geodesy review ROM SAF Report 14 Contents 1 Introduction 5 2 Gravity 6 2.1 Variation with latitude . .6 2.2 Variation with altitude . 10 3 Effective radius of Earth 11 4 Geopotential height 14 5 Radius of curvature 16 6 Undulation 20 7 Response to ROPP2.0 beta reviewer’s comments 22 8 Sensitivity studies 24 8.1 Sensitivity to surface gravity . 24 8.2 Sensitivity to effective radius . 27 8.3 Sensitivity to radius of curvature . 29 8.4 Sensitivity to undulation . 31 9 Summary and conclusions 33 10 Appendix: The geodesy of Rodworld 35 10.1 Intoduction . 35 10.2 Absence of rotation . 35 10.2.1 Geopotential . 36 10.2.2 Surface gravity . 36 10.2.3 Effective radius . 37 10.3 Presence of rotation . 40 10.3.1 Geopotential . 40 10.3.2 Surface gravity . 40 10.3.3 Effective radius . 41 10.4 Summary . 41 Bibliography 43 4 ROM SAF Report 14 Culverwell: ROPP geodesy review 1 Introduction Geodesy is the study of the measurement and representation of the Earth, including its gravitational field. A review of the geodesy calculations in ROPP was suggested by the ROPP2.0 beta reviewer ([1]). His concerns were that these calculations are not being done sufficiently accurately and/or consistently in ROPP, and he suggested that the treatment of geodetic terms should be re-examined. There were two main aspects to the reviewer’s concern: insufficient accuracy in the transformation between geometric and geopotential height, and the apparent inconsistency between the effective radius of the Earth and the radius of curvature of the Earth at the tangent point. Lewis ([6]) compared the early ROPP implementations of gravity, effective radius and geopotential, based on expressions in the Smithsonian Meteorological Tables ([14]), and versions which have since been implemented in ROPP, based on Mahoney’s ([8]) calculations using Somigliana’s equation. He found that the fractional differences between the calculations of gravity and of geopotential were negligible (< 10−5), but that the fractional difference in effective radius of 5 × 10−5 equated to 350 m, which was considered excessive. This is why ROPP has consistently used geodesy routines based on Somigliana’s equation since ROPP1.2. A follow-up note (Lewis 2010 pers. comm.) discussed the differences between these later expres- sions for gravity, effective radius and geopotential and those used in the Invert package, which is used to process ROM SAF NRT products. He found various inconsistencies between the two sets of expressions: a 3 × 10−5 fractional difference in the equatorial gravity, and up to 300 m difference in effective radius. The differences in geopotential height were generally less than 1 m, however, and therefore considered negligible. The treatment of the radius of curvature in ROPP has developed since ROPP2.0 was reviewed, and, as the reviewer recommended, expressions involving the true radius of curvature of the section of the ellipsoid intersected by the occultation plane are now used in the ropp_utils/coordinates routines that are used by the ropp_pp module of ROPP. This report attempts to tie all these threads together, and give a coherent account of the current geodesy calculations in ROPP. Throughout the following discussion of the wide range of available geodesy calculations it should be borne in mind that ROPP itself is internally consistent: all modules use common expressions for geodesic parameters and formulae which are held in the ropp_utils module. This is one of the strengths of the modular structure of ROPP. Geodesy is important to radio occultation in two distinct ways: “physically”, in connection with the effects of gravity on the atmosphere; and “mathematically”, in connection with the natural co-ordinate system in which to discuss the bending of radio signals during an occultation. These two facets will be discussed separately, before being brought together. 5 Culverwell: ROPP geodesy review ROM SAF Report 14 2 Gravity 2.1 Variation with latitude Numerous expressions for the Earth’s gravity field appear in the literature. The most authoritative reference appears to be the International Gravity Formula (Lambert 1945 [4]). Writing g(h;f) for the effective (ie including rotational effects) gravity at geometric height h and geographic (or geodetic) latitude f 1, Lambert gives the surface gravity as 2 2 Lambert: g(0;f) = geq(1 + b sin f − b1 sin 2f) (2.1) where (ignoring third order terms and some geometric shape factors like everyone else) b = (5=2)m − f − (17=14)m f ≈ 5:3 × 10−3 (2.2) and −6 b1 = ( f =8)( f + 2b) = ( f =8)(5m − f ) ≈ 5:9 × 10 : (2.3) Here, geq = g(0;0) is surface gravity at equator, f = (a − c)=a is the flattening of the oblate ellipsoid which is assumed to be described by the mean sea level geoid2 with semi-major and semi-minor axes 2 a and c respectively, and m = w a=geq is the ratio of centrifugal to gravitational force on the equator. For the Earth, not coincidentally3, f and m are numerically similar: f ≈ m ≈ 1=300. This makes b > 0, which implies (since b b1) that g increases towards the poles, as might be expected from both the reduced centrifugal “dilution” of gravity (m) and the smaller distance to the Earth’s centre ( f ). A glance at Eqn (2.2) shows that only one of these statements is true. More on this later. Numerically in Eqn (2.1): −2 −3 −6 geq = 9:7803253359 ms ; b = 5:3024396 × 10 ; b1 = 5:8496912 × 10 : (2.4) Eqn (2.1), or something very much like it, appears throughout the literature (eg [14], [8]). Somigliana’s equation for the surface gravity, as currently used in ROPP (ropp_utils/geodesy/gravity.f90), takes the form ([2]): q 2 2 2 Somigliana: g(0;f) = geq(1 + ks sin f)= 1 − e sin f (2.5) 1That is, the angle between the local vertical and the equatorial plane. (As opposed to the geocentric latitude, which is the angle between the radius vector from the Earth’s centre to the given point and the equatorial plane.) 2But see Section 6 for a discussion of the undulation. 3In Principia Mathematica, Newton derived f = 5m=4 for a uniformly rotating homogeneous blob of fluid (reported by White et al [16]). The non-uniform density of the Earth means that in practice f is rather closer to m. 6 ROM SAF Report 14 Culverwell: ROPP geodesy review where −3 ks(Somigliana’s constant) = (cgpo − ageq)=ageq ≈ 1:9 × 10 (2.6) and q e(eccentricity) = (a2 − c2)=a2 ≈ 8:2 × 10−2: (2.7) Here, gpo = g(0;p=2) is the surface gravity at the pole. In ROPP, the parameters in Eqn (2.5) take the values −2 −3 geq = 9:7803253359 ms ; ks = 1:931853 × 10 ; e = 0:081819: Li and Götze ([7]) quote the following second order expansion of Eqn (2.5): ∗ 2 2 Li and Götze: g(0;f) = geq(1 + f sin f − 1=4 f4 sin 2f) (2.8) where ∗ −3 f = (gpo − geq)=geq ≈ 5:3 × 10 (2.9) and −6 (1=4) f4 = ( f =8)(5m − f ) ≈ 5:9 × 10 : (2.10) [Aside: Eqn (2.10) is a bit deus ex machina as it stands since it requires something like Eqn (2.19) below to get the rotation rate m into it.